The probability density function of a continuous random variable 𝑥 is as follows: 𝑓 of 𝑥 is equal to two 𝑥 plus one over 18 if 𝑥 is greater than or equal to one and less than or equal to four and zero otherwise. Find the probability that 𝑥 is less than two and the probability that 𝑥 is greater than two but less than four.
For our continuous random variable 𝑥, the probability that 𝑥 takes a value in a particular interval can be found by finding the area below the graph of its probability density function 𝑓 of 𝑥 over that range of 𝑥-values. In this question, the probability density function is a linear function of 𝑥. And so, its graph will be a straight line. The probability that 𝑥 is less than two first of all then will be the area under this graph between the values of one and two. And we can see that this shape will be a trapezium. So we can work out its area using our formulae for the areas of 2D shapes.
In general, the area of a trapezium is found by finding half of the sum of the parallel sides which are often labelled as 𝑎 and 𝑏 and multiplying this by the perpendicular distance between them, the height of the trapezium. In our question, the height of this trapezium is the difference between the values of one and two. So ℎ is equal to one. The values of 𝑎 and 𝑏, the lengths of the two vertical sides of this trapezium, will be the values of the function 𝑓 of 𝑥 evaluated at one and two, respectively, 𝑓 of one and 𝑓 of two.
To find each of these values, we can substitute into the probability density function. 𝑓 of one is equal to two multiplied by one plus one over 18 which simplifies to three over 18. In the same way, 𝑓 of two is equal to two multiplied by two plus one over 18 which simplifies to five over 18. So we can now substitute into our formula for the area of a trapezium. The area is equal to one-half multiplied by three 18ths plus five 18ths multiplied by one. Three 18ths plus five 18ths is equal to eight 18ths. And we can then cancel a factor of two from the two in the denominator and the eight in the numerator, giving four 18ths. We can cancel a further factor of two to give two in the numerator and nine in the denominator. The area is therefore equal to two-ninths. Remember this area gives the probability that our continuous random variable 𝑥 is less than two. And so, we found that the probability that 𝑥 is less than two is equal to two-ninths.
For the second part of the question, we are asked to find the probability that 𝑥 is greater than two but less than four. So we’re looking for the area under this straight line between 𝑥-values of two and four. That’s the area that I’ve now shaded in pink. We could apply the same method that we just used to find the probability that 𝑥 is less than two by finding the area of this trapezium. But there is in fact an easier way.
For any continuous random variable, the sum of all probabilities must be equal to one. And therefore, the full area below the graph of its probability density function must also be equal to one. We can therefore find the probability that 𝑥 is greater than two, but less than four by subtracting the probability we’ve just found for the probability that 𝑥 is less than two from one. This gives one minus two-ninths which is equal to seven-ninths. You can of course confirm this by calculating the area of this trapezium if you wish.
Now, there is actually a more general method that we can use to find these areas. And it would certainly be essential to use this method if the graph of the probability density function was not a straight line. To find the area below a curve, we can use integration so the probability that 𝑥 is greater than two but less than four could be found by integrating the probability density function 𝑓 of 𝑥 with respect to 𝑥 between the limits of two and four. In the same way, the probability that 𝑥 is less than two could be found by integrating the probability density function with respect to 𝑥 between the limits of one and two. If you are familiar with integration, then you could perform this integration and confirm that it does indeed give the same answers.
We found using our method of the area of a trapezium that the probability 𝑥 is less than two is two-ninths and the probability that 𝑥 is greater than two but less than four is seven-ninths.